Mhaskar et al. [1]} study functions representable by hierarchical binary tree networks, comparing with shallow networks. Fig. REF  demonstrates shallow networks and hierarchical binary tree networks.
Denote by \(S_m\)  the class of shallow networks with \(m\)  neurons, in the form of
\(\nonumber x \rightarrow \sum _{k=1}^{m} a_k\sigma (w_k\cdot x+b_k)\) 
